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We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz-Minkowski space $\l^3=(\r^3,dx_1^2+dx_2^2-dx_3^2),$ with fundamental piece having a finite number $(n+1)$…
We show that a complete embedded maximal surface in the 3-dimensional Lorentz-Minkowski space $L^3$ with a finite number of singularities is, up to a Lorentzian isometry, an entire graph over any spacelike plane asymptotic to a vertical…
A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in $\l^3.$ In addition, $\sb$ is called of finite type if it…
Using techniques of integrable systems, we study a Weierstrass representation formula for timelike surfaces with prescribed mean curvature in Minkowski 3-space. It is shown that timelike minimal surfaces are obtained by integrating a pair…
We show that to every maximal surface with conelike singularities in Lorentz-Minkowski space $\mathbb{L}^3$ that can be locally represented as the graph of a smooth function, there exists a corresponding timelike minimal surface in…
We shall investigate maximal surfaces in Minkowski 3-space with singularities. Although the plane is the only complete maximal surface without singular points, there are many other complete maximal surfaces with singularities and we show…
The geometry and topology of complete nonorientable maximal surfaces with lightlike singularities in the Lorentz-Minkowski 3-space are studied. Some topological congruence formulae for surfaces of this kind are obtained. As a consequence,…
We show that the singularities of spacelike maximal surfaces in Lorentz-Minkowski 3-space generically consist of cuspidal edges, swallowtails and cuspidal cross caps. The same result holds for spacelike mean curvature one surfaces in de…
Until now, the only known maximal surfaces in Minkowski 3-space of finite topology with compact singular set and without branch points were either genus zero or genus one, or came from a correspondence with minimal surfaces in Euclidean…
We prove that any minimal (maximal) strongly regular surface in the three-dimensional Minkowski space locally admits canonical principal parameters. Using this result, we find a canonical representation of minimal strongly regular time-like…
We construct a one-parameter family of embedded doubly periodic minimal surfaces of genus three with four parallel ends. The Weierstrass data for each surface of the family are given and the two dimensional period problem is solved.
Bour's minimal surface has remarkable properties in three dimensional Minkowski space. We reveal the definite and indefinite cases of the Bour's surface using Weierstrass representations, and give some differential geometric properties of…
We construct Weierstrass data for higher genus embedded doubly periodic minimal surfaces and present numerical evidence that the associated period problem can be solved. In the orthogonal ends case, there previously was only one known…
We use the Bj\"orling problem in Lorentz-Minkowski space to obtain explicit parametrizations of maximal surfaces containing a circle and a helix. We investigate the Weierstrass representation of these surfaces.
We construct new integrable systems to present Weierstrass type representations for spacelike surfaces whose mean curvature vector $\mathbf{H}$ satisfies the null condition $\langle \mathbf{H}, \mathbf{H} \rangle=0$ in the four dimensional…
In this paper, we will give an Enneper-type representation for spacelike and timelike minimal surfaces in the Lorentz-Minkowski space $L^{3}$, using the complex and the paracomplex analysis (respectively). Then, we exhibit various examples…
We establish what semi-discrete linear Weingarten surfaces with Weierstrass-type representations in $3$-dimensional Riemannian and Lorentzian spaceforms are, confirming their required properties regarding curvatures and parallel surfaces,…
Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete…
A timelike minimal surface in Minkowski 3-space is a surface whose induced metric is Lorentzian and with vanishing mean curvature. Such surfaces have many kinds of singularities. In this paper, we prove existence and non-existence theorems…
The Weierstrass representation for minimal surfaces in $\mathbb{R}^3$ provides a flexible method for constructing minimal surfaces of arbitrary genus. The topological limitations of minimal surfaces interfere with this providing a more…